If the atom`(_100)Fm^(257)` follows the Bohr model the radius of `_(100)Fm^(257)` is `n` time the Bohr radius , then find `n` .

Options

A. 100
B. 200
C. 4
D. 1/4

Explanation

To solve the problem of finding the value of \( n \) for the atom \( _{100}Fm^{257} \) in the context of the Bohr model, we can follow these steps:

### Step 1: Understand the Bohr Model
The radius of an electron orbit in the Bohr model is given by the formula:
\[ R = \frac{n^2 R_0}{Z} \]
where:
- \( R \) is the radius of the orbit,
- \( n \) is the principal quantum number (the orbit number),
- \( R_0 \) is the Bohr radius (approximately \( 0.529 \, \text{Å} \)),
- \( Z \) is the atomic number (number of protons).

### Step 2: Identify the Atomic Number
For the atom \( _{100}Fm^{257} \):
- The atomic number \( Z = 100 \).

### Step 3: Determine the Electron Configuration
The electron configuration for \( Z = 100 \) (Fermium) is:
- The distribution of electrons in the shells is \( 2, 8, 18, 32, 50 \).
- This shows that the outermost electrons are in the fifth shell (or orbit), so \( n = 5 \).

### Step 4: Substitute Values into the Formula
Now, we substitute \( n = 5 \) and \( Z = 100 \) into the formula for the radius:
\[ R = \frac{5^2 R_0}{100} \]

### Step 5: Simplify the Expression
Calculating \( 5^2 \):
\[ 5^2 = 25 \]
Thus, we have:
\[ R = \frac{25 R_0}{100} = \frac{1}{4} R_0 \]

### Step 6: Relate to \( n \)
From the problem statement, we know that:
\[ R = n \cdot R_0 \]
Setting the two expressions for \( R \) equal gives:
\[ \frac{1}{4} R_0 = n R_0 \]
Dividing both sides by \( R_0 \) (assuming \( R_0 \neq 0 \)):
\[ n = \frac{1}{4} \]

### Final Answer
Thus, the value of \( n \) is:
\[ \boxed{\frac{1}{4}} \]